A NEW COARSELY RIGID CLASS OF BANACH SPACES

We prove that the class of reflexive asymptotic- $c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic- $c_{0}$ space $Y$ , then $X$ is also reflexive and asymptotic- $c_{0}$ . In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic- $c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.